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Tight bound of random interval packing

Part of: Limit theorems Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Wansoo T. Rhee
Wansoo T. Rhee*
Affiliation:
Ohio State University
*
∗ Postal address: Department of Management Science, The Ohio State University, Columbus, OH 43210, USA.
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Abstract

Consider n random intervals I 1, …, I N chosen by selecting endpoints independently from the uniform distribution. A packing of I 1, …, I N is a disjoint sub-collection of these intervals: its wasted space is the measure of the set of points not covered by the packing. We investigate the random variable W N equal to the smallest wasted space among all packings. Coffman, Poonen and Winkler proved that EW N is of order (log N)2/N. We provide a sharp estimate of log P(W N ≥ t (log N)2 / N) and log P(W N ≤ t (log N)2 / N) for all values of t.

Keywords

Interval packingwasted spaceuniform distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 90B35: Scheduling theory, deterministic

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 990 - 997
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported in part by NSF contract DMS-9303188.

References

Coffman, E. G. Jr, and Lueker, G. S. (1991). Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, New York.Google Scholar
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Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30.CrossRefGoogle Scholar
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